Vectors — Class 12 Maharashtra HSC Mathematics revision notes
A 7-page visual revision sheet for Vectors, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Vectors in one look
magnitude and direction together, added tip to tail, multiplied two different ways
Scalars, Vectors, and the Types That Matter
- a scalar needs only a magnitude, e.g. mass, length, time
- a vector needs BOTH magnitude and direction, e.g. displacement, velocity, force
- a vector AB is a directed segment: A the initial point, B the terminal point
- its magnitude is |AB|=l(AB), never negative
- the zero vector 0 has zero magnitude and no definite direction
- a unit vector has magnitude exactly 1, written a for the unit vector along a
- collinear vectors share a line of action, whatever their length or sense
- coplanar vectors lie in the same or in parallel planes
- every vector here is treated as FREE: it can be translated anywhere in space
- without changing its magnitude or direction
Four vectors that are easy to mix up
- Equal vectors: same magnitude AND same direction; positions can differ entirely
- Negative of a vector: same magnitude, OPPOSITE direction: if PQ=a then QP=-a
- Collinear vectors: same line of action only; can differ in magnitude or point opposite ways
- Co-initial vectors: share one starting point but generally point in different directions
Laws of Vector Addition
- commutative
a+b=b+a - associative
(a+b)+c=a+(b+c) - additive identity and inverse
a+0=a, a+(-a)=0 - triangle inequality
|a+b|≤|a|+|b|
Page 1 of 7. The full sheet, all 7 pages, opens free in the app.
What this Vectors sheet covers
- Scalars, Vectors, and the Types That Matter
- Four vectors that are easy to mix up
- Equal vectors
- Negative of a vector
- Collinear vectors
- Co-initial vectors
- Adding two vectors, two pictures of the same sum
- Laws of Vector Addition
- commutative
- associative
- additive identity and inverse
- triangle inequality
- Position Vectors and Points in Space
- Section Formula
- R divides AB internally in ratio m:n
- R divides AB externally in ratio m:n
- midpoint of AB, the case m=n
- centroid of a triangle A,B,C
- Special Cases: Coplanar, Collinear, Zero Vectors
- Section formula, internal and external
- Midpoint theorem, proved by vectors
- Diagonals of a parallelogram bisect each other
- Components, Direction Cosines and Direction Ratios
- The One Identity Every d.c. Triple Obeys
- direction cosines always satisfy
- unit vector written from its d.c.s
- d.c.s FROM d.r.s a,b,c, writing d=√a^2+b^2+c^2
- Direction cosines from components
- Scalar (Dot) Product: always a number
- Dot Product vs Cross Product
- Dot, a . b
- Cross, a x b
- Dot Product, in components
- for a=a_1i+a_2j+a_3k, b=b_1i+b_2j+b_3k
- a vector dotted with itself
- angle between two vectors
- scalar projection of b on a
- vector projection of b on a
- Dot product, angle between two vectors
- Vector (Cross) Product: always a vector
- Cross Product, in components, and area
- for a=(a_1,a_2,a_3), b=(b_1,b_2,b_3), the product is
- area of the parallelogram on a and b
- area of the triangle on a and b
- Cross product and area of a triangle
- Which product do you actually need?
- Scalar Triple Product (Box Product)
- Box Product, Volumes, and the Determinant
- for a,b,c with components a_i,b_i,c_i
- volume of the parallelepiped with coterminus edges a,b,c
- volume of the tetrahedron with coterminus edges a,b,c
- coplanarity test for a,b,c
- Volume of a parallelepiped, and a coplanarity check
- Symbols
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Vectors the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.